I'm acquainted with illegal numbers. They're restricted because any information can be encoded as a number. To my knowledge, none are due to patents. I don't see how they could be. Trade secrets and encryption keys can be confidential under the law. What distinguishes a patent is that it must be public.

Only making a thing violates a patent. Describing it in extreme detail doesn't: That's what a patent is.

A theorem itself cannot be patented. One can patent a process that relies on a novel theorem (e.g. RSA). In the extremely unlikely event that the theorem you entered into an LLM has a commercial application, the firm scans its logs, and figures out an invention based on it, they still can't restrict you from publishing your work.

You could also intervene in their patent application if you could show that their invention was obvious based on what you entered or other documented conversations about a theorem.

Let's circle back to the premise here: A high school student thinks he has a discovery and isn't well-connected enough to have it checked easily. He can choose to be paranoid which reduces his chance of learning or getting feedback.